Microsoft Word - numero_31_art_9 E.M. Nurullaev et alii, Frattura ed Integrità Strutturale, 31 (2015) 120-126; DOI: 10.3221/IGF-ESIS.31.09 120 Dependence of the mechanical fracture energy of the polymeric composite material from the mixture of filler fractions E. M. Nurullaev, A. S. Ermilov Perm National Research Polytechnic University, Perm, Russia. 614990 Perm, Komsomol prospect, 29 ergnur@mail.ru ABSTRACT. This paper for the first time presents an equation for calculating the mechanical fracture energy of the polymeric composite material (PCM) with regard to the basic formulation parameters. By means of the developed computer program the authors calculated the mechanical fracture energy of the polymer binder of the 3D cross-linked plasticized elastomer filled with multifractional silica. The solution of the integral equation was implemented using the corresponding dependence of stress on relative elongation at uniaxial tension. Engineering application of the theory was considered with respect to asphalt road covering. The authors proposed a generalized dependence of ruptural deformation of the polymer binder from the effective concentration of chemical and physical (intermolecular) bonds for calculating the mechanical fracture energy of available and advanced PCMs as filled elastomers. KEYWORDS. Energy failure; Elastomeric; Particulate filler; Binder; Polymer composite material. INTRODUCTION owadays, the current issue of technical chemistry is to ensure the required deformation and strength characteristics of advanced polymeric composite materials, in particular, 3D cross-linked filled plasticized elastomers. So far, to estimate the effect of the basic formulation parameters on the mechanical properties of such materials at uniaxial tension, Smith failure envelopes have been widely used [1-4]. However, this approach does not fully reflect the physical nature of the failure (or fracture) process. Therefore, it is interesting to derive an equation for calculating the mechanical fracture energy as a composite performance indicator of 3D cross-linked polymeric composite materials. The objective of this paper is a mathematical derivation of the equation for calculating the mechanical fracture energy of polymeric composite materials with regard to the basic formulation parameters by integrating the equation that describes the curve of uniaxial tension. In addition, by means of the computer program the authors calculated the mechanical fracture energy of a 3D cross- linked plasticized elastomer filled with multifractional silica. It is of practical interest for asphalt road covering as it greatly increases the service life of road pavements. N E.M. Nurullaev et alii, Frattura ed Integrità Strutturale, 31 (2015) 120-126; DOI: 10.3221/IGF-ESIS.31.09 121 THE THEORY OF CALCULATING THE MECHANICAL FRACTURE ENERGY echanical characteristics of polymeric composite materials (PCMs) based on the 3D cross-linked plasticized elastomeric matrix filled with solid silica particles significantly affect the service life of these materials. In this regard, the most important formulation (structural) parameters are the molecular structure of the polymeric matrix of the binder, its type and degree of plasticization, maximum volume filling, depending on the shape and fractional composition of dispersed filler particles, as well as the physical and chemical interaction at the binder-filler interface [5]. Direct and indirect optimization problems of developing new types of similar PCMs with the desired combination of stress-strain properties can be solved by means of a structural-mechanical dependence which was obtained earlier [6, 7]. Let us explore a variant of maintaining the integrity of РСМ until the sample breaks (Poisson's ratio → 0.5), which is of the most practical interest for increasing its service life, including working conditions of the rocket engine charge under domestic pressure. Earlier, in connection with PCMs, we presented a physical and mathematical description of the dependence of relative (related to the initial cross-section) stress (σ (MPa)) from stretch at break (αb(mm)) of the filled elastomer with account for its basic structural parameters [7]:      2 1 23 1 1 23   1 29 0.225 10 1 1.25 1 m ch r g m RT exp T T a                                   (1) where 3(  /  )ch c mol cm M   is the concentration of transverse chemical bonds in the polymer binder matrix; ρ (g/cm3) – density of the polymer binder;  cM – average internodal molecular weight of the 3D cross-linked polymer; φr (vol. fraction) = (1 – φsw) – polymer volume fraction in the binder; φsw (vol. fraction) – plasticizer volume fraction in the binder; R (J/K·mol) – universal gas constant; T∞ (К) – equilibrium temperature, at which intermolecular interaction (the concentration of transverse physical bonds – νph(mol/cm 3)) in the binder is negligible (νph (mol/cm3)→0); T (К)– sample test temperature; Tg (К)– structural glass-transition temperature of the polymer binder; a (с-1) – velocity shift ratio; φ (vol. fraction) – volume fraction of the dispersed filler; φm (vol. fraction) – maximum permissible volume filler fraction depending on the shape and fractional composition of the filler particles. In practice, we can estimate the structural parameter value cM by means of a molecular graph [7]:      32 3 32 2 21 1 12 2 23 3 322 2c nM f R f R f R f R f R f                    where R1 and R2 are molecular chains of two rubber substances with two kinds of reactive end groups f1 and f2; R3 – a molecule of the cross-linking agent with three antipodal reactive groups – f3; combinations of subscripts at f denote the chemical reaction products of i-th and j-th antipodal reactive end groups of R1-type and R2-type bifunctional polymers, as well as R3-type cross-linking trifunctional agent. Taking into account that usually R3<< R1 and R2, we can assume that cM is proportional to increment addition of the molar fraction of linear polymerization  1 12 2 nR f R   according to the molecular graph. The value of φm (vol. fraction) can be calculated [8] or determined by a viscosimetric method [9]. Relative elongation (α (mm)) is connected with deformation (ε (%)) by a well-known ratio rating: α = 1 + ε / 100. On conditions that the integrity of the material is maintained, true stress (σver (MPa)) is equal to the product of σ·α, but in practice it is more convenient to use conventional stress for comparison with research results of PCMs that do not remain integral until the sample breaks [7]. On the basis of the Eq. (1) we can write the following relation:        2 1 23 1 1 23   1 1 1 29 0.225 10      1 1.25 1 b b m ch r g m W d RT exp T T a                                           (2) where W (J),is energy (work) of mechanical fracture at uniaxial tension with dimensions: MPа  elongation (mm) = 1·103 J. The latter are the equivalent of the time during which the polymeric composite material resists increasing tensile stress. M E.M. Nurullaev et alii, Frattura ed Integrità Strutturale, 31 (2015) 120-126; DOI: 10.3221/IGF-ESIS.31.09 122 Сonsidering W (J,) and α (mm) to be variable and denoting the combination of the rest alphabetic and digital characters as constants C1 and C2 from the expression (2) we get the following: 2 1 3 1   1 1.25 1 m ch r m C RT                   23 1 2  29 0.225 10 gC exp T T a           which allows us to write the dependence (2) in compact form:   1 21 2 1 1 b W C C d         (3) The Eq. (3) can be solved by means of Table of Standard Integrals [10]: 2 3 1 1 2 1 1 2 1 1 1 1 b b b b W C d C C d C d C C d                   Further, as a result of integration and algebraic transformations, we obtain the equation: 3 3 2 1 1 2 2 3 2 2 3 1 2 2 b b b b b b W C C C                       (4) which leads, using the notations in (2), to the required dependence of the mechanical fracture energy of PCM on its basic formulation parameters:   2 3 3 2 1 23 1 3   3 2 2 3 1 1 1.25 29 exp  0.225 10   2 21 m b b b b ch r g b m W RT T T a                                                    (5) Let us note that the mechanical fracture energy (W (J)), is equal to zero when αb=1, indicating at its required normability. ULTIMATE ELONGATION AND ENERGY TO BREAK ltimate values of relative elongation (αb(mm)), as well as strain (εb(%)), can be estimated by considering the velocity and degree of strain of an average polymeric binder layer between the solid particles of the filler [7]: 3 3 0 0 3 3 0 3 1 1 1 f m m f b b m m f b b m a a                                     (6) where the indices «f» and "0" refer to the filled and free states of the 3D cross-linked plasticized polymer binder of PCM. For engineering use of the Eq. (5) when developing new PCMs based on 3D cross-linked plasticized elastomers it is necessary to know the value of the maximum relative elongation or ruptural deformation of the polymeric binder. As follows from the relation (1), the values of 0b (mm) or 0 b (%) are determined by the polymer volume fraction in the plasticized binder (φr (vol. fraction)), effective concentration of transverse bonds (νeff (mol/cm 3)), comprising permanent U E.M. Nurullaev et alii, Frattura ed Integrità Strutturale, 31 (2015) 120-126; DOI: 10.3221/IGF-ESIS.31.09 123 transverse chemical bonds (νch (mol/cm3)) and variable physical (intermolecular) bonds (νph (mol/cm3)), and the latter determine the temperature-velocity dependence of the mechanical characteristics:     1 1 233 3   1 1 29 0.225 10eff ch r ph g ch r gT T exp T T                    (7) The molecular structure parameter (the statistically average internodal molecular weight ( cM ) of 3D cross-linked systems based on low-molecular-weight polymers with terminal functional groups) was theoretically evaluated in the following paper [11]. However, the authors did not consider the molecular interaction, which as well as the mechanical properties depend, as was noted above, on a variety of factors [12-14]. Therefore, for use in engineering practice of determining the ruptural deformation of the free polymeric binder depending on the amount of νeff we have summarized the experimental data obtained earlier [7]. It turned out that the nonlinear experimental dependence    0 %b efff  for various polymeric binders, built on a logarithmic scale [7], is linearized in the coordinates: 0 0log log | cb b effM C      (8) where 0log | 3.1 cb M    corresponds to 0 | 1250% cb M    ; coefficient C = 40; 1  eff c d M   is in accordance with the formula (7). After algebraic transformations we obtain an empirical dependence: 3.1 400 10 effb     (9) MATERIAL AND METHODS reaking strain was measured on a tensile testing machine brand "Instron," at a rate of expansion "."The materials used in the study cross-linked elastomers based on viscous-flow low-molecular rubbers with terminal functional groups – poly(butyl formal sulfide), poly(ester urethane)hydroxide, polydiene epoxy urethane, poly(isoprene– butyl), carboxyl-terminated polybutadiene – cured by three functional agents with antipodal functional groups. Low-molecular rubbers (PDI-3B grade polydiene epoxy urethane with epoxy end groups and SKD-KTR grade carboxyl- terminated polybutadiene) were used as the polymer matrix. 3D cross-linking was performed using EET-1 grade epoxy resin. Mixtures of silica fractions with an average particle size (600 µm, 15 µm, and 1 µm) were used as the filler. The polymeric binder contained dibutyl phthalate as a plasticizer  [ 1 0.3]sw r    . Optimum values of the fraction parameters are listed in Tab. 1, 2 and 3. The selected standard relative strain rate is 1.4·10-3 c-1. Fig. 1 presents a generalized dependence of ruptural deformation of different polymer binders (on a logarithmic scale) on the square root of the effective concentration of transverse bonds (on a linear scale). Taking into account the relations (6), by means of the generalized dependence (9) we can determine elongation at break (  0 b mm ) of the polymeric binder in the free state and hence the ultimate elongation of the three-dimensionally cross- linked filled elastomer (  fb mm ), which allows us to calculate the energy of its mechanical fracture at uniaxial tension. The Eq. (5), showing the dependence of fracture energy on the parameter / m  , was applied for a PCM based on a low- molecular rubber. Fraction number Particle diameter, µm Pore volume ratio Optimum values of fraction volume ratio Maximum volume filling 1 15 0.386 0.2 0.84 2 600 0.244 0.8 Тable 1: Parameter values of mixtures of two silica fractions B E.M. Nurullaev et alii, Frattura ed Integrità Strutturale, 31 (2015) 120-126; DOI: 10.3221/IGF-ESIS.31.09 124 Fraction number Particle diameter, µm Pore volume ratio Optimum fraction volume ratio Maximum volume filling 1 1 0.465 0.05 0.94 2 15 0.386 0.149 3 600 0.244 0.801 Тable 2: Parameter values of mixtures of three silica fractions. Fraction number Particle diameter, µm Pore volume ratio Optimum fraction volume ratio Maximum volume filling 1 1 0.465 0.028 0.96 2 15 0.386 0.082 3 240 0.367 0.226 4 600 0.244 0.664 Table 3: Parameter values of mixtures of four silica fractions. Fraction quantity, piece 2, 3, 4 Experiment temperatures, К 223, 273, 323 Polymer glass transition temperature, К 175 Plasticizer glass transition temperature, К 185 Polymer volume-expansion coefficient 5·10-4 Plasticizer volume-expansion coefficient 7·10-4 Volume fraction of the polymer in the binder 0.25 Glass transition temperature of the composite elasomer, К 177 Filler volume ratio in mixed solid rocket propellants 0.75 Chemical bond concentration in the binder, mol/сm3 1·10-5 Maximum filler volume ratio 0.84; 0.94; 0.96 Тable 4: Initial data for calculating the mechanical fracture energy. For comparison, we considered composite materials based on polymeric binders with mixtures of two, three and four (Fig. 2) silica fractions. It is seen that, contrary to Smith failure envelopes [1-4] and [17], the mechanical fracture energy reflects the mechanical resistance of PCM as a filled elastomer more fully in the physical sense, which is important when estimating its operational suitability in particular materials. The dependencies in Fig. 2 allow to evaluate the influence of the quantity of fractions taken in the optimal ratio, on the amount of ruptural deformation (the value of the mechanical fracture energy being practically constant). For example, at the temperature of 223 K b (%) changes from 0 to 16% (2-fractional silica), from 0 to 25% (3-fractional silica), from 0 to 35% (4-fractional silica), which, respectively, leads to a double increase of b . A similar phenomenon is observed at temperatures of 273 K and 323 K. The latter circumstance is very favorable for the use of PCM as frost and waterproof asphalt coating of automobile roads. It is important to add that the increase of ruptural deformation of PCM as 3D cross-linked filled plasticized elastomer in accordance with the Eq. (1) is contributed by the decrease in the values of other structural parameters –  ,  ,  /ch r m    , – E.M. Nurullaev et alii, Frattura ed Integrità Strutturale, 31 (2015) 120-126; DOI: 10.3221/IGF-ESIS.31.09 125 as well as the structural glass-transition temperature of the polymer binder. In this case, the decrease in ultimate tensile stress (σb (MPa)), related to (αb (mm)) by the formula:   1 21 21b b b bC C       with previously adopted notations applied to the Eq. (1), occurs to a lesser degree. This can probably explain the corresponding increase in the values of W (J), at increasing εb(%). The theoretically derived dependence (5) of the mechanical fracture energy of a 3D cross-linked filled elastomer from the basic structural parameters of the composite can be recommended for solving direct and indirect problems when developing new PCMs as polymer composites for various purposes with the desired combination of performance characteristics [7]. In doing so, it is expedient to use computer programs, including mathematical optimization techniques [15, 16], which will help to reduce the development time and cost of raw materials, for example, when developing advanced PCMs. Figure 1: Experimental dependence    0 3% [ / ]b efff mol cm  for various polymeric binders based on: : Poly(butyl formal sulfide), : Poly(ester urethane) hydroxide, : Polydiene epoxy urethane, : Carboxyl-terminated polybutadiene, : Polyisoprene butyl; The data are given for standard conditions: Т=293 К and  =1.4·10-3 c-1. Figure 2: Dependence of the mechanical fracture energy on ruptural deformation of a PCM at temperatures: 1 – 223 K; 2 – 273 K; 3 – 323 K. : two-fraction composition; : three-fraction composition; : four-fraction composition; E.M. Nurullaev et alii, Frattura ed Integrità Strutturale, 31 (2015) 120-126; DOI: 10.3221/IGF-ESIS.31.09 126 CONCLUSION he article presented the equation linking the mechanical fracture energy of the 3D cross-linked polymer binder filled with multifractional silica and its basic formulation parameters. The authors proposed a generalized dependence of ruptural deformation of the polymer binder from the effective concentration of chemical and physical (intermolecular) bonds for calculating the mechanical fracture energy of new PCMs. It has been shown that the increase in the quantity of fractions of solid components taken in the optimum ratio leads to a double increase of ruptural deformation at constant mechanical fracture energy. REFERENCES [1] Smith, T. L., Symposium on stress-strain-time-temperature relationships in materials, Amer. Soc. Test. Mat. Spec. Publ., 325 (1962) 60-89. [2] Smith, T. L., Limited Characteristics of Cross-linked Polymers, J. Appl. Phys., 35 ( 1964) 27-32. [3] Smith, T. L., Relationship between the structure and tensile strength of elastomers, The mechanical properties of new materials (Translated from English by G.I. Barenblatta), Мir, (1966) 174-190. [4] Smith, T. L., Chy, W. H., Ultimate Tensile Properties of Elastomers, J. Polymer Sce., 10(1) (1972) 133-150. [5] Ermilov, A. S., Nurullaev, E. M., Mechanical Properties of Elastomers Filled with Solid Particles, Mechanics of Composite Materials, 48 (3)(2012) 243-252. [6] Ermilov, A. S., Nurullaev, E. M., Optimization of Fractional Composition of the Filler of Elastomer Composites, Mechanics of Composite Materials, 49 (3) (2013) 455-464. [7] Ermilov, A. S., Nurullaev, E. M., Mechanical Properties of Elastomers Filled with Solid Particles, Mechanics of Composite Materials, 48 (3) (2012) 1-14. [8] Ermilov, A. S., Fedoseev, A. M., Combinatorial-Multiplicative Method of Calculating the Limiting Filling of Composites with Solid Dispersed Components, Russian Journal of Applied Chemistry, 77(7) (2004) 1203-1205. [9] Ermilov, A. S., Nurullaev, E. M., Concentration dependence of reinforcing rubbers with dispersed fillers, Journal of Applied Chemistry, 85(8) (2012) 1371-1374. [10] Bronstein, I.N., Semendyaev, K.A., A guide to mathematics for engineers and students of engineering, Science, 544 (1986) [11] Zabrodin, V. B., Zykov, V. I., Chui, G. N., Molecular structure of cross-linked polymers, High-molecular compounds, XVII(1) (1975) 163-169. [12] Van Krevelen, D.W., Properties and the chemical structure of polymers, (Translated from English by A. Y. Malkina), Chemistry, (1976) 415. [13] Nielsen, L.E., Mechanical Properties of Polymers and Composites, (Translated from English by P.G. Babaevskyi), Chemistry, (1978) 311. [14] Manson, J.A., Sperling, L.H., Polymer Blends and Composites, (Translated from English by Y.K. Godovskyi), Chemistry, (1979) 440. [15] Certificate № 2012613349 RF, Software of identification and optimization of the packing density of solid dispersed fillers of polymer composite materials (Rheology), Ermilov A. S., Nurullaev E. M., Duregin К. А. – Priority of 09.04.2012. [16] Certificate № 2011615640 RF, Mathematical software of predicting physical and mechanical characteristics of filled elastomers. (Elastomer) / Ermilov A. S., Nurullaev E. M., Subbotina T.E., Duregin К. А. – Priority of 18.07.2011. [17] Ermilov, A. S., Nurullaev, E. M., Influence of formulation parameters on the mechanical failure energy of filled elastomers, Russian Journal of Applied Chemistry, 85(7) (2012) 1125 – 1127. T